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Model Uncertainty Quantification for Data Assimilation in partially observed Lorenz 96

Sahani Pathiraja, Peter Jan Van Leeuwen

Institut f¨ ur Mathematik Universit¨ at Potsdam

With thanks: Sebastian Reich, Georg Gottwald

Motivation

◮ Uncertainty Quantification important for successful DA ◮ Main focus: Ensemble Data Assimilation ◮ Model Uncertainty due to unresolved sub-grid scale processes

Problem Setting

System states xj evolve according to the following stochastic difference equation: xj = M(xj−1) + ηj ηj is an additive stochastic model error.

Problem Setting

System states xj evolve according to the following stochastic difference equation: xj = M(xj−1) + ηj ηj is an additive stochastic model error. Observations of the system are available in the form of: yj = Hxj + εj εj ∼ N(0, R)

Problem Setting

System states xj evolve according to the following stochastic difference equation: xj = M(xj−1) + ηj ηj is an additive stochastic model error. Observations of the system are available in the form of: yj = Hxj + εj εj ∼ N(0, R) Aim: estimate p(xj|yj) (i.e. filtering).

2 Layer Lorenz 96

Figure: Two Layer Lorenz 96 system with 8 coarse scale variables and 32 fine scale variables (taken from Arnold et al., 2013)

Aim

Build on existing stochastic and deterministic parameterization techniques: e.g. Wilks (2005), Crommelin & Vanden-Eijnden (2008), Kwasniok (2012), Arnold et al. (2013), Lu et al. (2017)

Aim

Build on existing stochastic and deterministic parameterization techniques: e.g. Wilks (2005), Crommelin & Vanden-Eijnden (2008), Kwasniok (2012), Arnold et al. (2013), Lu et al. (2017) Develop method to estimate distribution of η for the following conditions:

◮ No dynamical equations for the fine scale process ◮ Only partial observations of coarse scale process available

2 Layer Lorenz 96

dXk dt = −Xk−1(Xk−2 − Xk+1) − Xk + F + hx L

L

• l=1

Yl,k; k ∈ {1, ..., K} dYl,k dt = 1 ξ (−Yl+1,k(Yl+2,k − Yl−1,k) − Yl,k + hyXk; l ∈ {1, ..., L}

Assumptions

• 1. States are directly but partially observed (i.e. H is a

non-square (0, 1) matrix)

Assumptions

• 1. States are directly but partially observed (i.e. H is a

non-square (0, 1) matrix)

• 2. Model error ηj depends on some informative variable (e.g.

xj−1 or some reduced form of it)

Assumptions

• 1. States are directly but partially observed (i.e. H is a

non-square (0, 1) matrix)

• 2. Model error ηj depends on some informative variable (e.g.

xj−1 or some reduced form of it)

• 3. ||εj|| << ||ηj||

Assumptions

• 1. States are directly but partially observed (i.e. H is a

non-square (0, 1) matrix)

• 2. Model error ηj depends on some informative variable (e.g.

xj−1 or some reduced form of it)

• 3. ||εj|| << ||ηj||
• 4. Error statistics are the same at each point in time and space

(translation invariance): p(ηj[k]|xj−1) = p(ηb[l]|xb−1) ∀k, j, b, l

Proposed Method

Proposed Error Estimation Method

Proposed Error Estimation Method

ˆ ηo

1 = y1 − HM(ˆ

x0) ˆ η1 = HT ˆ ηo

1 + [H⊥]T ˆ

ηu

1

ˆ x1 = M(ˆ x0) + ˆ η1 ˆ ηo

2 = y2 − HM(ˆ

x1) . . .

Proposed Error Estimation Method

◮ Minimize total conditional variance:

• Var(ηj[k]|xj−1[k])dxj−1[k]

Proposed Error Estimation Method

◮ Minimize total conditional variance:

• Var(ηj[k]|xj−1[k])dxj−1[k]

≈

Nb

• i=1

SV (ηj[k]|xj−1[k] = xi−1) + SV (ηj[k]|xj−1[k] = xi) 2 ∆xi

Benchmark Method (B1)

◮ Analysis Increment Based Method

xfi

j = M(xai j−1) − αηi j

ηi

j ∼ N(bm, Pm)

Benchmark Method (B1)

◮ Analysis Increment Based Method

xfi

j = M(xai j−1) − αηi j

ηi

j ∼ N(bm, Pm)

where: α = tuning parameter bm = 1 N

N

• j=1

δxa

j

P= 1 N − 1

N

• j=1
• δxa

j − bm

δxa

j − bm

T δxa

j = 1

n

n

• i=1
• xai

j − xfi j

Benchmark Method (B2)

◮ Long Window Weak Constraint 4dVar based Method:

J(ηt:t+u) =

t+u

• j=t

ηT

j Qηj

+

t+u

• j=t

(Hxj − yj)TR−1(Hxj − yj) where: xj = M(xj−1) + ηj

◮ All other aspects same as proposed approach

Numerical Experiments - L96

Chaotic Two Layer Lorenz 96: dXk dt = −Xk−1(Xk−2 − Xk+1) − Xk + F + hx L

L

• l=1

Yl,k; k ∈ {1, ..., K} dYl,k dt = 1 ξ (−Yl+1,k(Yl+2,k − Yl−1,k) − Yl,k + hyXk; l ∈ {1, ..., L} Parameter Case Study 1 - large time scale sep. Case Study 2 - small time scale sep. ξ

1 128 ≈ 0.008

0.7 hx

• 0.8
• 2

hy 1 1 J 128 20 K 9 9 F 10 14

Numerical Experiments - L96

Observation Details:

◮ every 2nd Xk is measured ◮ 0.02 & 0.04 MTU for Case Study 1 and 2 respectively ◮ Negligible observation error: R = 10−7I

Error Estimation Results - Case Study 1

Error Estimation Results - Case Study 2

Why conditional variance minimization?

Figure: Snapshot of JQ values for method B2, proposed and true data for Case Study 2

Impact on Assimilation/Forecasts

Data Assimilation setup:

◮ Ensemble Transform Kalman Filter (ETKF)

(Wang & Bishop, 2004)

◮ ensemble size (n) = 1000 ◮ observation frequency - as per estimation period ◮ assimilation length - 3000 observation intervals

Forecast Skill

Non-Gaussian Uncertainties

Figure: Example Evolution of Forecast density in Case Study 1

Summary

◮ Proposed method for quantifying model uncertainty due to

unresolved sub-grid scale processes

◮ Difficult conditions: No dynamical equations of fine scale

process and partial observations of coarse scale process

◮ Improved representation of model uncertainty with minimal

prior knowledge

Further Work/Extensions

◮ Extension for including Obs Error ◮ Scalability

Acknowledgements

◮ This research has been partially funded by Deutsche

Forschungsgemeinschaft (DFG) through grant CRC 1294 ‘’Data Assimilation”

◮ We gratefully acknowledge Professor Georg Gottwald and

Professor Sebastian Reich for thoughtful discussions on this work.

References

◮ Arnold, H. M., Moroz, I. M. and Palmer, T. N. (2013) ‘Stochastic parametrizations and

model uncertainty in the Lorenz ’ 96 system’, Philosophical Transactions of the Royal Society A, 371. doi: dx.doi.org/10.1098/rsta.2011.0479.

◮ Crommelin, D. and Vanden-Eijnden, E. (2008) ‘Subgrid-Scale Parameterization with

Conditional Markov Chains’, Journal of the Atmospheric Sciences, 65(8), pp. 2661–2675. doi: 10.1175/2008JAS2566.1.

◮ Kwasniok, F. (2012) ‘Data-based stochastic subgrid-scale parametrization: an approach

using cluster-weighted modelling’, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370(1962), pp. 1061–1086. doi: 10.1098/rsta.2011.0384

◮ Lu, F., Tu, X., Chorin, A. J., Lu, F., Tu, X. and Chorin, A. J. (2017) ‘Accounting for model

error from unresolved scales in ensemble Kalman filters by stochastic parametrization’, Monthly Weather Review, p. MWR-D-16-0478.1. doi: 10.1175/MWR-D-16-0478.1.

◮ Mitchell, L. and Carrassi, A. (2015) ‘Accounting for model error due to unresolved scales

within ensemble Kalman filtering’, Quarterly Journal of the Royal Meteorological Society, 141(689), pp. 1417–1428. doi: 10.1002/qj.2451.

◮ Tremolet, Y. (2006) ‘Accounting for an imperfect model in 4D-Var’, Quarterly Journal of

the Royal Meteorological Society, 132(621), pp. 2483–2504. doi: 10.1256/qj.05.224.

◮ Wang, X., Bishop, C. H. and Julier, S. J. (2004) ‘Which Is Better, an Ensemble of

Positive-Negative Pairs or a Centered Spherical Simplex Ensemble’, Bulletin of the American Meteorological Society, pp. 2823–2829. doi: 10.1175/1520-0493(2004)132¡1590:WIBAEO¿2.0.CO;2.

◮ Wilks, D. S. (2005) ‘Effects of stochastic parameterizations in the Lorenz 96 system’,

Quarterly Journal of the Royal Meteorological Society, 131. doi: 10.1256/qj.04.03.